AppliedMath, Volume 3, Issue 2 (June 2023) – 12 articles

We have studied how the readability of a text can change in translation by considering Matthew’s Gospel, written in Greek, translated into Latin and 35 modern languages. We have found that the deep-language parameters $C_P$ (characters per word), $P_F$ (words per sentence), $I_P$ (words per interpunctions), $M_F$ (interpunctions per sentence) and a universal readability index $G_U$ of each translation are so diverse from language to language, and even within a given language for which there are many versions of Matthew—such as in English and Spanish—that the resulting texts mathematically seem to be diverse. The several tens of versions of Matthew’s Gospel studied appear to address very diverse audiences. If a reader could understand all of them well, he/she would have the impression of reading texts written by diverse authors, although all of them tell the same story. Full article

This research employs computational methods to analyze the velocity and mixture fraction distributions of a non-reacting Propane jet flow that is discharged into parallel co-flowing air under iso-thermal conditions. This study includes a comparison between the numerical results and experimental results obtained from the Sandia Laboratory (USA). The objective is to improve the understanding of flow structure and mixing mechanisms in situations where there is no involvement of chemical reactions or heat transfer. In this experiment, the Realizable k-ε eddy viscosity turbulence model with two equations was utilized to simulate turbulent flow on a nearly 2D plane (specifically, a 5-degree partition of the experimental cylinder domain). This was achieved using OpenFOAM open-source software and swak4Foam utility, with the reactingFoam solver being manipulated carefully. The selection of this turbulence model was based on its superior predictive capability for the spreading rate of both planar and round jets, as compared to other variants of the k-ε models. Numerical axial and radial profiles of different parameters were obtained for a mesh that is independent of the grid (mesh B). These profiles were then compared with experimental data to assess the accuracy of the numerical model. The parameters that are being referred to are mean velocities, turbulence kinetic energy, mean mixture fraction, mixture fraction half radius (L_f), and the mass flux diagram. The validity of the assumption that w߰ = v߰ for the determination of turbulence kinetic energy, k, seems to hold true in situations where experimental data is deficient in w߰. The simulations have successfully obtained the mean mixture fraction and its half radius, L_f, which is a measure of the jet’s width. These values were determined from radial profiles taken at specific locations along the X-axis, including x/D = 0, 4, 15, 30, and 50. The accuracy of the mean vertical velocity fields in the X-direction (U_mean) is noticeable, despite being less well-captured. The resolution of mean vertical velocity fields in the Y-direction (V_mean) is comparatively lower. The accuracy of turbulence kinetic energy (k) is moderate when it is within the range of U_mean and V_mean. The absence of empirical data for absolute pressure (p) is compensated by the provision of numerical pressure contours. Full article

An algebraic system is introduced which is very useful for performing scattering calculations in quantum field theory. It is the set of all real numbers greater than or equal to −m² with parity designation and a special rule for addition and subtraction, where m is the rest mass of the scattered particle. Full article

We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using the Wilks likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology was illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna, respectively, and also to the earlier (1998) Innsbruck experiment of Weihs et al. and the recent (2022) Munich experiment of Zhang et al., which investigates the use of a loophole-free Bell experiment as part of a protocol for device-independent quantum key distribution (DIQKD). Full article

Hepatocellular carcinoma (HCC) is the primary liver cancer that occurs the most frequently. The risk of developing HCC is highest in those with chronic liver diseases, such as cirrhosis brought on by hepatitis B or C infection and the most common type of liver cancer. Knowledge-based interpretations are essential for understanding the HCC microarray dataset due to its nature, which includes high dimensions and hidden biological information in genes. When analyzing gene expression data with many genes and few samples, the main problem is to separate disease-related information from a vast quantity of redundant gene expression data and their noise. Clinicians are interested in identifying the specific genes responsible for HCC in individual patients. These responsible genes may differ between patients, leading to variability in gene selection. Moreover, ML approaches, such as classification algorithms, are similar to black boxes, and it is important to interpret the ML model outcomes. In this paper, we use a reliable pipeline to determine important genes for discovering HCC from microarray analysis. We eliminate redundant and unnecessary genes through gene selection using principal component analysis (PCA). Moreover, we detect responsible genes with the random forest algorithm through variable importance ranking calculated from the Gini index. Classification algorithms, such as random forest (RF), naïve Bayes classifier (NBC), logistic regression, and k-nearest neighbor (kNN) are used to classify HCC from responsible genes. However, classification algorithms produce outcomes based on selected genes for a large group of patients rather than for specific patients. Thus, we apply the local interpretable model-agnostic explanations (LIME) method to uncover the AI-generated forecasts as well as recommendations for patient-specific responsible genes. Moreover, we show our pathway analysis and a dendrogram of the pathway through hierarchical clustering of the responsible genes. There are 16 responsible genes found using the Gini index, and CCT3 and KPNA2 show the highest mean decrease in Gini values. Among four classification algorithms, random forest showed $96.53\%$ accuracy with a precision of $97.30\%$. Five-fold cross-validation was used in order to collect multiple estimates and assess the variability for the RF model with a mean ROC of $0.95\pm 0.2$. LIME outcomes were interpreted for two random patients with positive and negative effects. Therefore, we identified 16 responsible genes that can be used to improve HCC diagnosis or treatment. The proposed framework using machine-learning-classification algorithms with the LIME method can be applied to find responsible genes to diagnose and treat HCC patients. Full article

In the first part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude. Full article

In this work, we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes a two-step iteration without surd or complex numbers. The computational test we performed reveals that such a generalization improves the accuracy in computation of the arctangent function by many orders of magnitude with increasing integer M, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of $\pi$ in the Machin-like formulas. Full article

Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of ${10}^{-16}$, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals. Full article

Our research involves analyzing the latest models used for electricity price forecasting, which include both traditional inferential statistical methods and newer deep learning techniques. Through our analysis of historical data and the use of multiple weekday dummies, we have proposed an innovative solution for forecasting electricity spot prices. This solution involves breaking down the spot price series into two components: a seasonal trend component and a stochastic component. By utilizing this approach, we are able to provide highly accurate predictions for all considered time frames. Full article

In interval-valued three-way decision, the reflection of decision-makers’ preference under the full consideration of interval-valued characteristics is particularly important. In this paper, we propose an interval-valued three-way decision model based on the cumulative prospect theory. First, by means of the interval distance measurement method, the loss function and the gain function are constructed to reflect the differences of interval radius and expectation simultaneously. Second, combined with the reference point, the prospect value function is utilized to reflect decision-makers’ different risk preferences for gains and losses. Third, the calculation method of cumulative prospect value for taking action is given through the transformation of the prospect value function and cumulative weight function. Then, the new decision rules are deduced based on the principle of maximizing the cumulative prospect value. Finally, in order to verify the effectiveness and feasibility of the algorithm, the prospect value for decision-making and threshold changes are analyzed under different risk attitudes and different radii of the interval-valued decision model. In addition, compared with the interval-valued decision rough set model, our method in this paper has better decision prospects. Full article

In this work, we studied convergence rates using quotient convergence factors and root convergence factors, as described by Ortega and Rheinboldt, for Hestenes’ Gram–Schmidt conjugate direction method without derivatives. We performed computations in order to make a comparison between this conjugate direction method, for minimizing a nonquadratic function f, and Newton’s method, for solving $\nabla f=0$. Our primary purpose was to implement Hestenes’ CGS method with no derivatives and determine convergence rates. Full article